Normalized defining polynomial
\( x^{16} - 3 x^{15} + 4 x^{14} - 6 x^{13} + 6 x^{12} - 24 x^{11} + 85 x^{10} - 150 x^{9} + 222 x^{8} + \cdots + 1 \)
Invariants
Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[6, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-298892216010498046875\) \(\medspace = -\,3^{8}\cdot 5^{12}\cdot 179\cdot 1021^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(19.04\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
Galois root discriminant: | $3^{1/2}5^{3/4}179^{1/2}1021^{1/2}\approx 2475.868173489506$ | ||
Ramified primes: | \(3\), \(5\), \(179\), \(1021\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-179}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{36\!\cdots\!51}a^{15}+\frac{211089986662298}{36\!\cdots\!51}a^{14}+\frac{504178652109895}{36\!\cdots\!51}a^{13}+\frac{11\!\cdots\!38}{36\!\cdots\!51}a^{12}+\frac{33219222813734}{36\!\cdots\!51}a^{11}+\frac{7367533910326}{124344021508219}a^{10}-\frac{11\!\cdots\!47}{36\!\cdots\!51}a^{9}-\frac{76034443725204}{36\!\cdots\!51}a^{8}-\frac{676168841620307}{36\!\cdots\!51}a^{7}-\frac{13\!\cdots\!91}{36\!\cdots\!51}a^{6}-\frac{372587161822864}{36\!\cdots\!51}a^{5}-\frac{726804838981338}{36\!\cdots\!51}a^{4}+\frac{956114934557104}{36\!\cdots\!51}a^{3}+\frac{795389941812920}{36\!\cdots\!51}a^{2}+\frac{204163705521251}{36\!\cdots\!51}a-\frac{11\!\cdots\!95}{36\!\cdots\!51}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $10$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( -1 \) (order $2$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
Fundamental units: | $\frac{4022301831}{74502109951}a^{15}-\frac{13107236664}{74502109951}a^{14}+\frac{14295431796}{74502109951}a^{13}-\frac{20027749792}{74502109951}a^{12}+\frac{22451196414}{74502109951}a^{11}-\frac{84582957921}{74502109951}a^{10}+\frac{362435389632}{74502109951}a^{9}-\frac{583502230599}{74502109951}a^{8}+\frac{774841242087}{74502109951}a^{7}-\frac{1504647556963}{74502109951}a^{6}+\frac{1801752058449}{74502109951}a^{5}-\frac{496479255087}{74502109951}a^{4}+\frac{315197053596}{74502109951}a^{3}-\frac{908007903018}{74502109951}a^{2}+\frac{201668925633}{74502109951}a+\frac{109865346368}{74502109951}$, $\frac{694342154987377}{36\!\cdots\!51}a^{15}+\frac{363532244669148}{36\!\cdots\!51}a^{14}-\frac{21\!\cdots\!27}{36\!\cdots\!51}a^{13}+\frac{483527895377125}{36\!\cdots\!51}a^{12}-\frac{53\!\cdots\!80}{36\!\cdots\!51}a^{11}-\frac{413238935736524}{124344021508219}a^{10}+\frac{58\!\cdots\!66}{36\!\cdots\!51}a^{9}+\frac{50\!\cdots\!71}{36\!\cdots\!51}a^{8}-\frac{53\!\cdots\!63}{36\!\cdots\!51}a^{7}+\frac{39\!\cdots\!76}{36\!\cdots\!51}a^{6}-\frac{30\!\cdots\!52}{36\!\cdots\!51}a^{5}+\frac{31\!\cdots\!38}{36\!\cdots\!51}a^{4}+\frac{63\!\cdots\!48}{36\!\cdots\!51}a^{3}+\frac{13\!\cdots\!18}{36\!\cdots\!51}a^{2}-\frac{16\!\cdots\!36}{36\!\cdots\!51}a+\frac{14\!\cdots\!23}{36\!\cdots\!51}$, $\frac{890970453271146}{36\!\cdots\!51}a^{15}-\frac{14\!\cdots\!38}{36\!\cdots\!51}a^{14}+\frac{10\!\cdots\!87}{36\!\cdots\!51}a^{13}-\frac{28\!\cdots\!84}{36\!\cdots\!51}a^{12}+\frac{486925831088251}{36\!\cdots\!51}a^{11}-\frac{642193333893853}{124344021508219}a^{10}+\frac{49\!\cdots\!59}{36\!\cdots\!51}a^{9}-\frac{54\!\cdots\!00}{36\!\cdots\!51}a^{8}+\frac{88\!\cdots\!85}{36\!\cdots\!51}a^{7}-\frac{19\!\cdots\!97}{36\!\cdots\!51}a^{6}+\frac{10\!\cdots\!05}{36\!\cdots\!51}a^{5}+\frac{27\!\cdots\!00}{36\!\cdots\!51}a^{4}+\frac{95\!\cdots\!65}{36\!\cdots\!51}a^{3}-\frac{58\!\cdots\!12}{36\!\cdots\!51}a^{2}-\frac{23\!\cdots\!23}{36\!\cdots\!51}a+\frac{13\!\cdots\!34}{36\!\cdots\!51}$, $\frac{889025585909608}{36\!\cdots\!51}a^{15}-\frac{270871117105116}{36\!\cdots\!51}a^{14}-\frac{14\!\cdots\!31}{36\!\cdots\!51}a^{13}-\frac{485835222305467}{36\!\cdots\!51}a^{12}-\frac{42\!\cdots\!66}{36\!\cdots\!51}a^{11}-\frac{554407892506673}{124344021508219}a^{10}+\frac{23\!\cdots\!98}{36\!\cdots\!51}a^{9}+\frac{21\!\cdots\!72}{36\!\cdots\!51}a^{8}-\frac{15\!\cdots\!76}{36\!\cdots\!51}a^{7}-\frac{33\!\cdots\!87}{36\!\cdots\!51}a^{6}-\frac{22\!\cdots\!03}{36\!\cdots\!51}a^{5}+\frac{29\!\cdots\!51}{36\!\cdots\!51}a^{4}+\frac{21\!\cdots\!44}{36\!\cdots\!51}a^{3}+\frac{93\!\cdots\!00}{36\!\cdots\!51}a^{2}-\frac{15\!\cdots\!03}{36\!\cdots\!51}a+\frac{20\!\cdots\!91}{36\!\cdots\!51}$, $\frac{578055038697441}{36\!\cdots\!51}a^{15}-\frac{558986029965648}{36\!\cdots\!51}a^{14}-\frac{106841111663471}{36\!\cdots\!51}a^{13}-\frac{10\!\cdots\!34}{36\!\cdots\!51}a^{12}-\frac{12\!\cdots\!22}{36\!\cdots\!51}a^{11}-\frac{392924703937624}{124344021508219}a^{10}+\frac{23\!\cdots\!97}{36\!\cdots\!51}a^{9}-\frac{11\!\cdots\!28}{36\!\cdots\!51}a^{8}+\frac{23\!\cdots\!51}{36\!\cdots\!51}a^{7}-\frac{74\!\cdots\!78}{36\!\cdots\!51}a^{6}-\frac{36\!\cdots\!59}{36\!\cdots\!51}a^{5}+\frac{98\!\cdots\!68}{36\!\cdots\!51}a^{4}+\frac{38\!\cdots\!64}{36\!\cdots\!51}a^{3}+\frac{34\!\cdots\!17}{36\!\cdots\!51}a^{2}-\frac{50\!\cdots\!60}{36\!\cdots\!51}a+\frac{21\!\cdots\!29}{36\!\cdots\!51}$, $\frac{750285338554635}{36\!\cdots\!51}a^{15}-\frac{35\!\cdots\!75}{36\!\cdots\!51}a^{14}+\frac{58\!\cdots\!87}{36\!\cdots\!51}a^{13}-\frac{72\!\cdots\!47}{36\!\cdots\!51}a^{12}+\frac{97\!\cdots\!16}{36\!\cdots\!51}a^{11}-\frac{730206100989594}{124344021508219}a^{10}+\frac{91\!\cdots\!32}{36\!\cdots\!51}a^{9}-\frac{19\!\cdots\!38}{36\!\cdots\!51}a^{8}+\frac{28\!\cdots\!39}{36\!\cdots\!51}a^{7}-\frac{48\!\cdots\!84}{36\!\cdots\!51}a^{6}+\frac{73\!\cdots\!00}{36\!\cdots\!51}a^{5}-\frac{53\!\cdots\!79}{36\!\cdots\!51}a^{4}+\frac{22\!\cdots\!12}{36\!\cdots\!51}a^{3}-\frac{32\!\cdots\!92}{36\!\cdots\!51}a^{2}+\frac{24\!\cdots\!77}{36\!\cdots\!51}a-\frac{33\!\cdots\!96}{36\!\cdots\!51}$, $\frac{40517508011687}{36\!\cdots\!51}a^{15}-\frac{128004638010026}{36\!\cdots\!51}a^{14}+\frac{140536890975151}{36\!\cdots\!51}a^{13}-\frac{197731810492582}{36\!\cdots\!51}a^{12}+\frac{221929507108975}{36\!\cdots\!51}a^{11}-\frac{30539149616646}{124344021508219}a^{10}+\frac{35\!\cdots\!34}{36\!\cdots\!51}a^{9}-\frac{57\!\cdots\!84}{36\!\cdots\!51}a^{8}+\frac{76\!\cdots\!12}{36\!\cdots\!51}a^{7}-\frac{14\!\cdots\!84}{36\!\cdots\!51}a^{6}+\frac{18\!\cdots\!06}{36\!\cdots\!51}a^{5}-\frac{49\!\cdots\!68}{36\!\cdots\!51}a^{4}+\frac{28\!\cdots\!58}{36\!\cdots\!51}a^{3}-\frac{92\!\cdots\!21}{36\!\cdots\!51}a^{2}+\frac{20\!\cdots\!35}{36\!\cdots\!51}a+\frac{28\!\cdots\!29}{36\!\cdots\!51}$, $\frac{28\!\cdots\!64}{36\!\cdots\!51}a^{15}-\frac{76\!\cdots\!60}{36\!\cdots\!51}a^{14}+\frac{93\!\cdots\!00}{36\!\cdots\!51}a^{13}-\frac{14\!\cdots\!61}{36\!\cdots\!51}a^{12}+\frac{13\!\cdots\!69}{36\!\cdots\!51}a^{11}-\frac{22\!\cdots\!46}{124344021508219}a^{10}+\frac{22\!\cdots\!83}{36\!\cdots\!51}a^{9}-\frac{36\!\cdots\!18}{36\!\cdots\!51}a^{8}+\frac{53\!\cdots\!98}{36\!\cdots\!51}a^{7}-\frac{10\!\cdots\!59}{36\!\cdots\!51}a^{6}+\frac{11\!\cdots\!95}{36\!\cdots\!51}a^{5}-\frac{54\!\cdots\!93}{36\!\cdots\!51}a^{4}+\frac{43\!\cdots\!42}{36\!\cdots\!51}a^{3}-\frac{52\!\cdots\!81}{36\!\cdots\!51}a^{2}+\frac{22\!\cdots\!46}{36\!\cdots\!51}a-\frac{18\!\cdots\!84}{36\!\cdots\!51}$, $\frac{14\!\cdots\!94}{36\!\cdots\!51}a^{15}-\frac{42\!\cdots\!49}{36\!\cdots\!51}a^{14}+\frac{53\!\cdots\!53}{36\!\cdots\!51}a^{13}-\frac{81\!\cdots\!86}{36\!\cdots\!51}a^{12}+\frac{78\!\cdots\!88}{36\!\cdots\!51}a^{11}-\frac{11\!\cdots\!96}{124344021508219}a^{10}+\frac{12\!\cdots\!50}{36\!\cdots\!51}a^{9}-\frac{20\!\cdots\!34}{36\!\cdots\!51}a^{8}+\frac{30\!\cdots\!96}{36\!\cdots\!51}a^{7}-\frac{55\!\cdots\!72}{36\!\cdots\!51}a^{6}+\frac{65\!\cdots\!42}{36\!\cdots\!51}a^{5}-\frac{33\!\cdots\!00}{36\!\cdots\!51}a^{4}+\frac{22\!\cdots\!56}{36\!\cdots\!51}a^{3}-\frac{30\!\cdots\!95}{36\!\cdots\!51}a^{2}+\frac{15\!\cdots\!14}{36\!\cdots\!51}a-\frac{13\!\cdots\!84}{36\!\cdots\!51}$, $\frac{18\!\cdots\!14}{36\!\cdots\!51}a^{15}-\frac{35\!\cdots\!48}{36\!\cdots\!51}a^{14}+\frac{29\!\cdots\!58}{36\!\cdots\!51}a^{13}-\frac{66\!\cdots\!01}{36\!\cdots\!51}a^{12}+\frac{28\!\cdots\!09}{36\!\cdots\!51}a^{11}-\frac{13\!\cdots\!62}{124344021508219}a^{10}+\frac{11\!\cdots\!29}{36\!\cdots\!51}a^{9}-\frac{14\!\cdots\!57}{36\!\cdots\!51}a^{8}+\frac{21\!\cdots\!90}{36\!\cdots\!51}a^{7}-\frac{46\!\cdots\!64}{36\!\cdots\!51}a^{6}+\frac{33\!\cdots\!81}{36\!\cdots\!51}a^{5}-\frac{17\!\cdots\!87}{36\!\cdots\!51}a^{4}+\frac{20\!\cdots\!16}{36\!\cdots\!51}a^{3}-\frac{17\!\cdots\!72}{36\!\cdots\!51}a^{2}-\frac{22\!\cdots\!75}{36\!\cdots\!51}a+\frac{23\!\cdots\!16}{36\!\cdots\!51}$ (assuming GRH) | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 10347.5753866 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{5}\cdot 10347.5753866 \cdot 1}{2\cdot\sqrt{298892216010498046875}}\cr\approx \mathstrut & 0.187555850906 \end{aligned}\] (assuming GRH)
Galois group
$C_4^4.C_2\wr C_4$ (as 16T1771):
A solvable group of order 16384 |
The 190 conjugacy class representatives for $C_4^4.C_2\wr C_4$ |
Character table for $C_4^4.C_2\wr C_4$ |
Intermediate fields
\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\), 8.8.1292203125.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 16 siblings: | data not computed |
Degree 32 siblings: | data not computed |
Minimal sibling: | 16.4.52401279790283203125.2 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | $16$ | R | R | $16$ | ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | $16$ | ${\href{/padicField/41.4.0.1}{4} }^{3}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | $16$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{5}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
$p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
---|---|---|---|---|---|---|---|
\(3\) | 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
\(5\) | 5.16.12.1 | $x^{16} + 16 x^{14} + 16 x^{13} + 124 x^{12} + 192 x^{11} + 368 x^{10} - 16 x^{9} + 1462 x^{8} + 2688 x^{7} + 2544 x^{6} + 5232 x^{5} + 14108 x^{4} + 4800 x^{3} + 8592 x^{2} - 6512 x + 13041$ | $4$ | $4$ | $12$ | $C_4^2$ | $[\ ]_{4}^{4}$ |
\(179\) | 179.2.0.1 | $x^{2} + 172 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
179.2.0.1 | $x^{2} + 172 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
179.2.0.1 | $x^{2} + 172 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
179.2.0.1 | $x^{2} + 172 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
179.2.1.2 | $x^{2} + 179$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
179.2.0.1 | $x^{2} + 172 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
179.4.0.1 | $x^{4} + x^{2} + 109 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(1021\) | $\Q_{1021}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{1021}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $4$ | $2$ | $2$ | $2$ |